Clearly a spoof on Problem 1, 3rd Romanian Selection Test 2007. Such functions $f$ are in one-to-one correspondence with chains $\emptyset = X_0\subsetneq X_1 \subsetneq \cdots \subsetneq X_m \subseteq X$, for some $0\leq m \leq \min\{n, k-1\}$, and values $1\leq f(X_0) < f(X_1) < \cdots < f(X_m) \leq k$, which (uniquely) determine $f(S) = \max\{f(X_i) \mid 0\leq i \leq m, \, X_i \subseteq S\}$ for all $S\in P$. Actual counting becomes a straightforward affair.